A quadratic upper bound on the size of a synchronizing word in one-cluster automata

International audienceČerný's conjecture asserts the existence of a synchronizing word of length at most (n-1)² for any synchronized n-state deterministic automaton. We prove a quadratic upper bound on the length of a synchronizing word for any synchronized n-state deterministic automaton satisfying the following additional property: there is a letter a such that for any pair of states p, q, one has p*ar = q*as for some integers r, s (for a state p and a word w, we denote by p*w the state reached from p by the path labeled w). As a consequence, we show that for any finite synchronized prefix code with an n-state decoder, there is a synchronizing word of length O(n²). This applies in particular to Huffman codes

On the Number of Synchronizing Colorings of Digraphs

We deal with $k$-out-regular directed multigraphs with loops (called simply \emph{digraphs}). The edges of such a digraph can be colored by elements of some fixed $k$-element set in such a way that outgoing edges of every vertex have different colors. Such a coloring corresponds naturally to an automaton. The road coloring theorem states that every primitive digraph has a synchronizing coloring. In the present paper we study how many synchronizing colorings can exist for a digraph with $n$ vertices. We performed an extensive experimental investigation of digraphs with small number of vertices. This was done by using our dedicated algorithm exhaustively enumerating all small digraphs. We also present a series of digraphs whose fraction of synchronizing colorings is equal to $1-1/k^d$, for every $d \ge 1$ and the number of vertices large enough. On the basis of our results we state several conjectures and open problems. In particular, we conjecture that $1-1/k$ is the smallest possible fraction of synchronizing colorings, except for a single exceptional example on 6 vertices for $k=2$.Comment: CIAA 2015. The final publication is available at http://link.springer.com/chapter/10.1007/978-3-319-22360-5_1

A Quadratic Upper Bound on the Size of a Synchronizing Word in One-Cluster Automata

Černý's conjecture asserts the existence of a synchronizing word of length at most (n - 1)2 for any synchronized n-state deterministic automaton. We prove a quadratic upper bound on the length of a synchronizing word for any synchronized n-state deterministic automaton satisfying the following additional property: there is a letter a such that for any pair of states p, q, one has p·ar = q·as for some integers r, s (for a state p and a word w, we denote by p·w the state reached from p by the path labeled w). As a consequence, we show that for any finite synchronized prefix code with an n-state decoder, there is a synchronizing word of length O(n2). This applies in particular to Huffman codes. © 2011 World Scientific Publishing Company

‘Left behind places’: a geographical etymology

‘Left behind places’ has become the leitmotif of geographical inequalities since the 2008 crisis. Yet, the term’s origins, definition and implications are poorly specified and risk obscuring the differentiated problems and pathways of different kinds of areas. This paper explicates the geographical etymology and spatial imaginary of ‘left behind places’. It argues that the appellation and its spatial expression have modified how geographical inequalities are understood and addressed by recovering a more relational understanding of multiple ‘left behind’ conditions, widening the analytical frame beyond only economic concerns, and opening up interpretations of the ‘development’ of ‘left behind places’ and their predicaments and prospects. While renewing interest in fundamental urban and regional concerns, what needs to endure from the ascendance of the ‘left behind places’ label is the terminology and spatial imaginary of reducing geographical inequalities and enhancing social and spatial justice

Effective-field-theory approach to persistent currents

Using an effective-field-theory (nonlinear sigma model) description of interacting electrons in a disordered metal ring enclosing magnetic flux, we calculate the moments of the persistent current distribution, in terms of interacting Goldstone modes (diffusons and cooperons). At the lowest or Gaussian order we reproduce well-known results for the average current and its variance that were originally obtained using diagrammatic perturbation theory. At this level of approximation the current distribution can be shown to be strictly Gaussian. The nonlinear sigma model provides a systematic way of calculating higher-order contributions to the current moments. An explicit calculation for the average current of the first term beyond Gaussian order shows that it is small compared to the Gaussian result; an order-of-magnitude estimation indicates that the same is true for all higher-order contributions to the average current and its variance. We therefore conclude that the experimentally observed magnitude of persistent currents cannot be explained in terms of interacting diffusons and cooperons.Comment: 12 pages, no figures, final version as publishe

Odd Frequency Pairing in the Kondo Lattice

We discuss the possibility that heavy fermion superconductors involve odd-frequency pairing of the kind first considered by Berezinskii. Using a toy model for odd frequency triplet pairing in the Kondo lattice we are able to examine key properties of this new type of paired state. To make progress treating the strong $n_f=1$ constraint in the Kondo lattice model we use the technical trick of a Majorana representation of the local moments, which permits variational treatments of the model without a Gutzwiller approximation. The simplest mean field theory involves the development of bound states between the local moments and conduction electrons, characterized by a spinor order parameter. We show that this state is a stable realization of odd frequency triplet superconductivity with surfaces of gapless excitations whose spin and charge coherence factors vanish linearly in the quasiparticle energy. A $T^3$ NMR relaxation rate coexists with a linear specific heat. We discuss possible extensions of our toy model to describe heavy fermion superconductivity.Comment: 67 page

Critical temperature of an anisotropic superconductor containing both nonmagnetic and magnetic impurities

The combined effect of both nonmagnetic and magnetic impurities on the superconducting transition temperature is studied theoretically within the BCS model. An expression for the critical temperature as a function of potential and spin-flip scattering rates is derived for a two-dimensional superconductor with arbitrary in-plane anisotropy of the superconducting order parameter, ranging from isotropic s-wave to d-wave (or any pairing state with nonzero angular momentum) and including anisotropic s-wave and mixed (d+s)-wave as particular cases. This expression generalizes the well-known Abrikosov-Gor'kov formula for the critical temperature of impure superconductors. The effect of defects and impurities in high temperature superconductors is discussed.Comment: 4 eps figure

Strongly coupled quantum criticality with a Fermi surface in two dimensions: fractionalization of spin and charge collective modes

We describe two dimensional models with a metallic Fermi surface which display quantum phase transitions controlled by strongly interacting critical field theories below their upper critical dimension. The primary examples involve transitions with a topological order parameter associated with dislocations in collinear spin density wave ("stripe") correlations: the gapping of the order parameter fluctuations leads to a fractionalization of spin and charge collective modes, and this transition has been proposed as a candidate for the cuprates near optimal doping. The coupling between the order parameter and long-wavelength volume and shape deformations of the Fermi surface is analyzed by the renormalization group, and a runaway flow to a non-perturbative regime is found in most cases. A phenomenological scaling analysis of simple observable properties of possible second order quantum critical points is presented, with results quite similar to those near quantum spin glass transitions and to phenomenological forms proposed by Schroeder et al. (cond-mat/0011002).Comment: 16 pages, 4 figures; (v2) additional clarifying remark

The in-plane paraconductivity in La_{2-x}Sr_xCuO_4 thin film superconductors at high reduced-temperatures: Independence of the normal-state pseudogap

The in-plane resistivity has been measured in $La_{2-x}Sr_xCuO_4$ (LSxCO) superconducting thin films of underdoped ($x=0.10,0.12$), optimally-doped ($x=0.15$) and overdoped ($x=0.20,0.25$) compositions. These films were grown on (100)SrTiO$_3$ substrates, and have about 150 nm thickness. The in-plane conductivity induced by superconducting fluctuations above the superconducting transition (the so-called in-plane paraconductivity, $\Delta\sigma_{ab}$) was extracted from these data in the reduced-temperature range 10^{-2}\lsim\epsilon\equiv\ln(T/\Tc)\lsim1. Such a $\Delta\sigma_{ab}(\epsilon)$ was then analyzed in terms of the mean-field--like Gaussian-Ginzburg-Landau (GGL) approach extended to the high-$\epsilon$ region by means of the introduction of a total-energy cutoff, which takes into account both the kinetic energy and the quantum localization energy of each fluctuating mode. Our results strongly suggest that at all temperatures above Tc, including the high reduced-temperature region, the doping mainly affects in LSxCO thin films the normal-state properties and that its influence on the superconducting fluctuations is relatively moderate: Even in the high-$\epsilon$ region, the in-plane paraconductivity is found to be independent of the opening of a pseudogap in the normal state of the underdoped films.Comment: 35 pages including 10 figures and 1 tabl